One remark that Penelope Maddy makes several times in Naturalism in Mathematics, is that if the indispensability argument was really important in justifying mathematics, then set theorists should be looking to debates over quantum gravity to settle questions of new axioms. Since this doesn’t seem to be happening, she infers that the indispensability argument can’t play the role Quine and Putnam (and perhaps her earlier book?) argued that it does.
However, since the awarding of the Fields Medal to the physicist Ed Witten in 1990, it’s not totally clear that Maddy is right about this. Set theorists certainly don’t pay much attention to string theory and related theories, but other mathematicians in low-dimensional topology and algebraic geometry seem to. I don’t know much about the details, but from what I understand, physicists have conjectured some deep and interesting connections between seemingly disparate areas of mathematics, in order to explain (or predict?) particular physical phenomena. These connections have rarely been rigorously proved, but they have stimulated mathematical research both in pursuing the analogies and attempting to prove them. Although the mathematicians often find the physicists’ work frustratingly imprecise and non-rigorous, once the analogies and connections have been suggested by physicists, mathematicians get very interested as well.
If hypothetically, one of these connections was to turn out to be independent of ZFC, I could imagine that there would at least be a certain camp among mathematicians that would take this as evidence for whatever large cardinal (or other) principle was needed to prove the connection. Set theorists themselves haven’t paid too much attention to these issues, because the interesting connections are in mathematical areas traditionally considered quite distant from set theory. Instead, they have traditionally looked at intra-set-theoretic considerations to justify large cardinals. But if it became plausible that some of these other debates would turn out to be connected, I’m sure they would start paying attention to the physics research, contrary to what Maddy suggests.
Antimeta 
If string theory uses any ‘foundational’ mathematics, it’s going to be higher-dimensional category theory (e.g., here). It’s intriguing to see a split opening up between ‘foundational’ theories which relate closely to ‘mainstream’ mathematics (and so presumably have at least a chance of being applied), and those which don’t. The model theorist Angus MacIntyre has a very interesting paper on this, and discusses how little contact Godel makes with the mainstream (abstract).
We were discussing something along these lines at this post. I think philosophers could do an important job by probing this distinction. Is there any good reason that so much of the mathematical knowledge of philosophers lies in disconnected foundational theories?
This strikes me as a Very Good Point. I guess the next relevant question to ask is: is it at all reasonable (or even conceivable) that any mathematical claim that these physicists are making could end up being independent of ZFC? This is a sort of unreasonable question to ask, since it appears to ask us to predict the scientific future, but I would be very happy to have even just a toy example in which a claim independent of ZFC ends up playing some plausible role in a physical theory…
Greg, we have comments addressing your question over here. Naturally, it all turns out to be very delicate what it means to say a physical theory uses a particular axiom.
David —
Thanks for the link to an interesting discussion. I looked through the whole thing, and my mathematical capabilities are pretty low, but I actually didn’t see anything that addressed my question (or even really Kenny’s question). I saw plenty of examples of how physical theories do rely on the axiom of choice, but no examples of how physical theories might rely on axioms beyond ZFC. (Though again, it’s very possible that somewhere in that ZFC-bashing the answer was there, and went way over my mathematical head.)
Greg,
Perhaps this was a little indirect, but the Axiom of Universes was discussed. You might argue that if a physical theory used a high dose of category theory, and some have been formulated so to do, it could be said to be relying upon this axiom, which is outside of ZFC.